Wave filter



H. w.. BopE WAVE FILTER May 21, 1935.

Filed June 7, 1955 2 Sheets-Sheet 1 w w a .95 FREQUENCY RA 770 FIG. 4

//v vs/vro/e H. W. 8005 w fiw bw ATTORNEY Patented r, 21'", 1935 I;

2,002,216 PATENT orrlci:

' emu wave rmrnn UNITED sTAT 1:-:s'

Hendrik W. Bode, New York. Bell Telephone laboratories,

N. Y, minor to Incorporated,

New York. N. Y a corporation of New York Application June i, 1933, Serial No. 674,619

'1 Claim. (Cl. 178-44) attenuation peaks.

In broad-band filters of the ladder, or seriesshunt type, the introduction of peaks into the attenuation characteristic is usually eflected by the inclusion in the filter of one or'more of the, so-called M-derived sections described in U. S.

Patent 1,538,964 issued May 26, 1925, to O. J.

Zobel. With no dissipation in the filters elements, the attenuation at these peaks would be infinite but in practical filters the natural dissipation-limits the' attenuation to a finite value. As the peaks are brought closer to the transmission band limits for the purpose of sharpening the cut-off; the eifect of the natural dissipation in diminishing the attenuation becomes more and more marked so that it becomes ime practicable to provide peaks closer to the cut-oil than about five per cent of the cut-oil frequency.

In accordancewith the present invention, the dissipation in a shunt resonant branch of a filter is compensated by means of a resistance bridged across adjacent series branches, thus permitting the shunt branch to introduce an infinite attenuation at its resonance frequency. The network thus formed retains the general characteristics of the original filter and can be connected in tandem with other sections without the introduction of noticeable reflection eflects. 'Approximate neutralization of. the shunt resistance extends over a range of frequencies on both sides of the peak and when the peak is located close to a cutofi frequency, the neutralization extends into the transmission band with a resultant diminution of the attenuation close to the cut-off.

By means of the invention, it has been found possible to produce infinite attenuation peaks at frequencies clolerto the cut-off than one per cent of-thecut-oi'l frequency and at the same time to improve the uniformity of the transmission inthe band up to points very close to the band limits.

The nature of the invention will be more fully understood from the following detailed description and by reference to the attached'drawingl, p v

' where h denotes the filter cut-oi! frequency, 1,,

of which:

Fig. l shows'in schematic form one embodiment of the invention;

Fig. 2 is a schematic explanatory of the network of Fig. l; Fig. 3 illustrates the characteristics of the network'of Pig. 1; 1 Fig. 4 shows a modified form of the invention;

' Fig. 5 is a schematic explanatory of the network of Fig. 4;

Figs. 6 and 7 illustrate additional networks embodying the invention;

Fig. 8 shows a modified form of the network of Fig. 4, and

Figs. 9 and 10 show respectively a further form of the invention and its prototype filter.

The network shown schematically in Fig. 1.

represents a low-pass filter section in accordance with the invention. It comprises a symmetrical T- network made up of two equal series inductances designated hand a shunt branch consisting of an inductance L: in series with a capacity C: together with a relatively high resistance 2R bridging the two series inductances. The T- network is a mid-series terminated filter section of the so-called M-derived type described in the above mentioned U. S. patent to Zobel, the shunt branch being proportioned to resonate at approximately the frequency of the attenuation peak.

At the resonance frequency the impedance of the shunt branch is very low and would be zero except for the effect of dissipation, represented mainly by the resistance of the inductance In.

In many cases, the resistance of this coil may be v quite low, giving a very high peak of attenuation, but when it is attempted to bring the peak very close to the cut-off of the filter the inductance L2 must be made very large with the result that its resistance is high and the shunting effect of the branch at resonance is diminished.

Design formula! for most of the important types of M-derlved filter sections are given in an article by A. J. Zobel on the Theory and Design of Uniform and Composite Electric Wave Filters, Bell System Technical Journal, vol. II, No. 1, January 1923. Fora low-pass filter of the type corresponding to the T-network oi Fig. 1 having series inductances- L0 and shunt inductance and capacity L03 and C02 respectively, the element values are given by a i es the frequency of the attenuation peak, and K0 the zero frequency value of the characteristic im-' which shows that the shunt inductance becomes very large relatively to the series inductance as that the cut-01f frequency and the peak frequency may be accurately placed.

The determination of the proper value of the bridging resistance and the necessary modification of the element values will be readily understood from a consideration of the network shown in Fig. 2 which is the T equivalent of the network of Fig. 1. The shunt branch of this network includes, in addition to the original elements L2 and C2, an inductance Lu: and a negative impedance consisting of an inductance --L1 2 and a resistance -R/2. The series branches each consist of an inductance L shunted by a resistance R. The network of Fig. 2 is not intended to represent a physically realizable equivalent of the network of the invention shown in Fig. 1, the negative elements being, in general impossible to construct except as rough approximations. It is, however, a mathematical equivalent which serves to illustrate in a simple and graphic manner the relationships which should obtain amongst the elements of the network of Fig. 1. The equivalence of the two networks may be shown by deriving the lattice equivalent network for each by means of the bisection theorem of A. C. Bartlett, described in an article entitled An Extension of a Property of Artificial Lines, Phil. Mag. (London), vol. 4, No. 24, Nov. 1927.

It is evident that the negative impedance appearing in the shunt branch may be proportioned to have an effective resistance which just neutralizes the total positive resistance of the branch and if this neutralization is made to oc-- cur at the frequency for which the total reactance is'also zero, the attenuation of the network will be infinite at this frequency.

When the attenuation peak is located very close to the cut-off the series inductance will in stantially wholly due to the resistance of the inductance coil L: which will be designated 1'2. It follows from Equation (3) and from the composition of the shunt branch that the neutralization of this resistance requires R. m m, rm

where w is the value of a corresponding to the peak frequency. The requirement that the re-.

proper location of the cut-off frequency with respect to the peak, particularly when the two are close together, and also that the filter will have substantially the same general transmission characteristics as the M-derived section. These requirements, expressed mathematically, are

From Equations (4) to (7) inclusive, the required values of L1, L2, and R are found to be the followin The following example will illustrate the application of the design formula: given above. The design data are as follows: Cut-off frequency 2500 cycles per second; characteristicimpedance at zero frequency 800 ohms; attenuation peak to be located at a frequency one hall of one per cent above the cut-off frequency. Inductance coils having a ratio of reactance to resistance of 200 will be assumed.

From equations, the values of L01, C02, and L02 are found to be Lo1=.005l henry Lo2=.252 henry Cn2=.0159 X 10- farads The resistance of the inductance L0: is 19.8 ohms at the frequency of infinite attenuation. Since the final value of inductance L2 is very close to L02, the value of resistance 12 may be taken at 19.8 ohms. The values of 2R, L1 and L: are then found by means of Equations 8 to be 2R=405 ohms, L1=.00627 henry, L=.251 henry.

The capacity C: is the same as C02.

The attenuation of this network is illustrated by curve ID of Fig. 3 which shows the attenuation loss in decibels plotted as ordinates against the frequency ratio f/fc as abscissa?" For pur, pose of comparison dotted curve II shows the loss of an ordinary M-derived filter section having the same cut-oft and attenuation peak frequencies. Curve I0 shows a greater loss at frequencies below the cut-off, this being due to the presence of the bridging resistance, but a markedly sharper cutoff and a much higher attenuation at the peak are also indicated.

A modified form of the invention having substantially the same transmission characteristics as the network of Fig. 1, but having series and shunt inductances of more nearly the same order' placed by a smaller inductance L: shunted by a capacity Ca. I

This'network is a true low-pass filter having only a single attenuating range and a single transmission range if the combination LaCa is proportioned to be anti-resonant at the same frequency as the combination of L1, and 02/2 in parallel. However, since in the cases primarily of interest, when the attenuation peak and .the cut-off are locatedvery close together, the latter anti-resonance is at so high a frequency as to be well out out the useful frequency range, it is not of importance that the two anti-resonances should be made to coincide.

From a comparison of the equivalent, T-networks shown in Figs. 2 and it will be seen that in the shunt branch of the latter the inductance L1/2 is absent and in each of the branches the parallel combinations of L1 and R are shunted by the capacity C2. Since the combination of L1 and 02/2 in parallel is anti-resonant at a very high frequency the capacity may be ignored at the frequencies of interest. Moreover, since the inductance 1.1/2 in the case of Fig. 2 is a very small part of the total shunt inductance its omission does not noticeably effect the network. Correction factors may, of course, be used in the design formula to take account of these modifications.

The combination La and C: should be so proportioned that the effective inductance at the peak frequency is approximately the same as the shunt inductance L2 of Fig. 1. More accurately, it should be proportioned to make the. effective inductance of the shunt branch of the equivalent T-network equal to the inductance L02 .as given by Equation (1) at the peak frequency. At frequencies below anti-resonance the reactance of a parallel connected inductance and capacity, such as. La and C3, is of an inductive character and is greater than the reactance of the inductance alone. At any given frequency in this range the combination may, therefore,

have a relatively large effective inductance although the inductance element itself may be relatively small. The above requirement of making the effective inductance of the combination La C3 equal to the inductance L02 at the peak frequency may, therefore, be met using a relatively small value for the inductance L3.

The general principle of neutralizingthe dissipation in a shunt resonant circuit by means of a series bridging resistance may be applied to high-pass'and band-pass filters as well as to' low-pass filters. Fig. 6 shows a high-pass filter in accordance with the invention comprising a T-network having series capacities C1, a shunt branch consisting of an inductance L2 and a capacity C: in series, together with a bridging resistance 2R. A band-pass filter is shown in Fig. '7 which differs from the filters of Figs. 1 and 6 only in that the series branches of the T consist of resonant combinations L1 and C1 instead of simple inductances or capacities.

The T-networks in each of these filters approxi- 'mate to known types of M-derived filters described in thearticle by Zobel herelnbefore menw tioned. The modification factors for the element values to take account of the bridging resistance are determined by the same general procedure as outlined above. j

A second modification of the network of Fig.

1 is illustrated by Fig. ii. The network is the same ,7, as that of Fig. 4 exceptthat the parallel induct ance and resistance 2L1 and IR in the bridging branch of Fig. 4 are replaced by an inductance and resistance 2L1 and IR. in series. The ele ments 2L1 and 2R are so proportioned that the impedance of their series combination at the frequency of infinite attenuation is the same as that of the bridging branch of, Fig. 4. The transmission characteristic of the network near the cut-ofi and the peak of infinite attenuation, then, is substantially the same as that of the structure of Fig. 4. In the network'of Fig. 8, however, the line current ows through the resistance 2R at low frequencies so that the network gives some transmission loss in this range. This transmission loss is, in fact, nearly the same as that at the cut-off. The transmission loss of the structure of Fig. 8 is thus substantially distortionless throughout the entire transmission band, which is a great advantage in'many ",the above mentioned paper by Zobel. For example, Fig. 10 shows a mid-shunt M-derived lowpass section. In the absence of dissipation the.

anti-resonant network inthe series branch has infinite impedance, and gives infinite attenuation, at its frequency of anti-resonance. When dissipation is considered, however, the impedance of the series branch is finite at all frequencies ,and' only a finite peak of attenuation is secured.

In accordance with the invention, the effects of dissipation can be balanced out by a low. resistance in the shunt branch, of a bridged-T networkwhose series and bridging impedances are respectively the shunt and series impedances of the original filter section. The configuration is shown by Fig. 9. It can be derived from the structure of Fig. 1 by interchanging the bridging'and shunt impedances of that figure and then replacing each impedance of the structure by its inverse network. Thus; for example, Ca of Fig.

a- 2 ."c.. c.'" where K0 is, as before, the characteristic impedance of the network at zero frequency. It will be understood that the same process of interchanging the shunt and bridging impedances of the bridged-T and then, replacing each branch by its inverse network can also be applied to all of the other structures herein described.

What is claimed is:

1. A wave, filternetwork comprising three impedance branches disposed in series-shunt relation between a pair of input terminals and a pair of output terminals, said branch impedances consisting of reactance elements proportioned with 'respect to each other to provide a broad transmission band, and one of said branches including an inductance coil and a capacity adapted by their resonance characteristic to produce an attenuation peak at afrequency close to the edge of said transmission band, and a resistance included in a fourth branch of the network proportioned to compensate the resistance of saidinductance coil at the frequency of the attenuation peak whereby the attenuation is made 5 bstantially infinite at the peak frequency.

tenuation peak whereby the attenuation at the peak frequency is made substantially infinite.

3. A wave filter section comprising two similar series impedances and a resonant shunt impedance connected to form a symmetrical T-networkbetween a pair of input terminals and a pair of output terminals, and a resistance bridging said two series impedances, the reactances of said impedances and the resonance frequency of said shunt impedance being proportioned with respect to each other to provide a transmission band and an attenuation peak close to the band, andsaid bridging resistance being proportioned-to cmpensate the energy dissipation in said shunt impedance at the resonance frequency thereof whereby the attenuation at the peak frequency is made substantially infinite.

4. A wave filter section in accordance with claim 3 in which the shunt impedance is proportioned to resonate at a frequency differing from a cut-off frequency of the filter by less than one v per cent.

5. A wave filter section in accordance with claim Bin which the series branches of the T-net- Work are constituted by equal inductances and the shunt branch by an inductance and 2. ca-

pacity in series, whereby a low-pass transmission band is defined, and in which the shunt branch is proportioned to resonate at a frequency less than one per cent above the cut-off frequency of the band.

6. A wave filter section in accordance with claim 3 in which the series branches of the T-network are constituted by equal capacities and the shunt branch by an inductance and a capacity in series, whereby a high-pass transmission band is defined, and in which the shunt branch is resonant at a frequency less than one per cent below the cut-ofl frequency of the band.

'7. A wave filter section comprising two equal capacities and a shunt impedance constituted by an inductance and a capacity in parallel arranged as a symmetrical T-network between a pair of input terminalsand a pair of output terminals, an inductance connected to bridge said equal capacities, said inductance and the elements of said T-network being'proportioned to provide a low-pass transmission band and an attenuation peak at a frequency close to the cut-off of said band, and a resistance connected in parallel with 

